QFunC Mixed Model

\[\begin{align} \quad R &= \begin{pmatrix} 1 & \rho_{12} & \cdots & \rho_{1J} \\ \rho_{12} & 1 & \ddots & \cdots \\ \vdots & \ddots & \ddots & \ddots \\ \rho_{J1} & \vdots & \ddots & 1 \end{pmatrix}, \Omega = SRS, \quad S = \operatorname{diag}(k_{\eta_1}^{1/2}, \dotsc, k_{\eta_J}^{1/2}) \\ \label{eq:varMats} \operatorname{Var}(\epsilon) &= \Sigma_\epsilon := \operatorname{diag}(\sigma^2_{\epsilon_1} I_{ML_1},\dotsc , \sigma^2_{\epsilon_J} I_{ML_J}) \\ \operatorname{Var}(\gamma) &= \Sigma_\epsilon^{1/2}\Lambda \Sigma_\epsilon^{1/2}, \quad \Lambda = \operatorname{diag}(C_1 \otimes B_1,\dotsc, C_J \otimes B_J) \\ \operatorname{Var}(\eta) &= (\tilde{\Sigma}_\epsilon^{1/2}\Omega \tilde{\Sigma}_\epsilon^{1/2})\otimes A \end{align}\]

Stage 1 parameters: \((k_{\gamma_j}, \sigma^2_{\gamma_j}, \tau_{\gamma_j}, \phi_j, \sigma^2_{\epsilon_j})\)

Stage 2 parameters: \((\rho_{jj'}, k_{\eta_j}, k_{\eta_{j'}}, \tau_\eta, \sigma^2_{\eta})\)

Need to choose suitable parameters for our simulations.

• Large \(\delta_j\) means strong common signal relative to voxel-level signal. Easier to estimate \(\rho\).
• For fixed \(\delta_j\), larger \(\psi_j\) means stronger spatial signal.

In simulated data, we choose parameters such that \(\delta_j = 0.1, 0.5, 0.7\) and \(\psi_j = 0.2, 0.5, 0.8\).

• Spatial misspecification: \(C_j\) [Castruccio et al. (2018)]
  • Anisotropic Matérn
  • Partition region \(j\) into two subregions with centroids \(v_{j1}^*, v_{j2}^*\).
  • Consider \(R = \operatorname{diag}(1, 1.2, 1.5)\).
  • \([C_j]_{\ell\ell'} = \sum_{s=1}^2 w_{s\ell} w_{s\ell'} K(\big\lVert{R(v_{j\ell} - v_{j\ell'})}\big\rVert; \nu_j, \phi_j)\)
  • \(w_{sl} = (\big\lVert{v_{j\ell} - v_{js^*}}\big\rVert)^{-1}\) is the inverse distance to the centroid.